2isqrt (example 3.6)

Percentage Accurate: 37.9% → 99.7%

Time: 14.1s

Alternatives: 6

Speedup: 2.0×

Specification

\[x > 1 \land x < 10^{+308}\]

Math

\[\begin{array}{l} \\ \frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- (/ 1.0 (sqrt x)) (/ 1.0 (sqrt (+ x 1.0)))))

C

double code(double x) {
	return (1.0 / sqrt(x)) - (1.0 / sqrt((x + 1.0)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.0d0 / sqrt(x)) - (1.0d0 / sqrt((x + 1.0d0)))
end function

Java

public static double code(double x) {
	return (1.0 / Math.sqrt(x)) - (1.0 / Math.sqrt((x + 1.0)));
}

Python

def code(x):
	return (1.0 / math.sqrt(x)) - (1.0 / math.sqrt((x + 1.0)))

Julia

function code(x)
	return Float64(Float64(1.0 / sqrt(x)) - Float64(1.0 / sqrt(Float64(x + 1.0))))
end

MATLAB

function tmp = code(x)
	tmp = (1.0 / sqrt(x)) - (1.0 / sqrt((x + 1.0)));
end

Wolfram

code[x_] := N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[(1.0 / N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 37.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- (/ 1.0 (sqrt x)) (/ 1.0 (sqrt (+ x 1.0)))))

C

double code(double x) {
	return (1.0 / sqrt(x)) - (1.0 / sqrt((x + 1.0)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.0d0 / sqrt(x)) - (1.0d0 / sqrt((x + 1.0d0)))
end function

Java

public static double code(double x) {
	return (1.0 / Math.sqrt(x)) - (1.0 / Math.sqrt((x + 1.0)));
}

Python

def code(x):
	return (1.0 / math.sqrt(x)) - (1.0 / math.sqrt((x + 1.0)))

Julia

function code(x)
	return Float64(Float64(1.0 / sqrt(x)) - Float64(1.0 / sqrt(Float64(x + 1.0))))
end

MATLAB

function tmp = code(x)
	tmp = (1.0 / sqrt(x)) - (1.0 / sqrt((x + 1.0)));
end

Wolfram

code[x_] := N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[(1.0 / N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{x + 1}} \leq 0:\\ \;\;\;\;\frac{-0.5 \cdot {x}^{-0.5} + -0.625 \cdot \frac{{x}^{-0.5}}{x}}{-x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(x + 1\right) - x}{x \cdot \left(x + 1\right)}}{{x}^{-0.5} + {\left(x + 1\right)}^{-0.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (/ 1.0 (sqrt x)) (/ -1.0 (sqrt (+ x 1.0)))) 0.0)
   (/ (+ (* -0.5 (pow x -0.5)) (* -0.625 (/ (pow x -0.5) x))) (- x))
   (/
    (/ (- (+ x 1.0) x) (* x (+ x 1.0)))
    (+ (pow x -0.5) (pow (+ x 1.0) -0.5)))))

C

double code(double x) {
	double tmp;
	if (((1.0 / sqrt(x)) + (-1.0 / sqrt((x + 1.0)))) <= 0.0) {
		tmp = ((-0.5 * pow(x, -0.5)) + (-0.625 * (pow(x, -0.5) / x))) / -x;
	} else {
		tmp = (((x + 1.0) - x) / (x * (x + 1.0))) / (pow(x, -0.5) + pow((x + 1.0), -0.5));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (((1.0d0 / sqrt(x)) + ((-1.0d0) / sqrt((x + 1.0d0)))) <= 0.0d0) then
        tmp = (((-0.5d0) * (x ** (-0.5d0))) + ((-0.625d0) * ((x ** (-0.5d0)) / x))) / -x
    else
        tmp = (((x + 1.0d0) - x) / (x * (x + 1.0d0))) / ((x ** (-0.5d0)) + ((x + 1.0d0) ** (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (((1.0 / Math.sqrt(x)) + (-1.0 / Math.sqrt((x + 1.0)))) <= 0.0) {
		tmp = ((-0.5 * Math.pow(x, -0.5)) + (-0.625 * (Math.pow(x, -0.5) / x))) / -x;
	} else {
		tmp = (((x + 1.0) - x) / (x * (x + 1.0))) / (Math.pow(x, -0.5) + Math.pow((x + 1.0), -0.5));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if ((1.0 / math.sqrt(x)) + (-1.0 / math.sqrt((x + 1.0)))) <= 0.0:
		tmp = ((-0.5 * math.pow(x, -0.5)) + (-0.625 * (math.pow(x, -0.5) / x))) / -x
	else:
		tmp = (((x + 1.0) - x) / (x * (x + 1.0))) / (math.pow(x, -0.5) + math.pow((x + 1.0), -0.5))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(x)) + Float64(-1.0 / sqrt(Float64(x + 1.0)))) <= 0.0)
		tmp = Float64(Float64(Float64(-0.5 * (x ^ -0.5)) + Float64(-0.625 * Float64((x ^ -0.5) / x))) / Float64(-x));
	else
		tmp = Float64(Float64(Float64(Float64(x + 1.0) - x) / Float64(x * Float64(x + 1.0))) / Float64((x ^ -0.5) + (Float64(x + 1.0) ^ -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (((1.0 / sqrt(x)) + (-1.0 / sqrt((x + 1.0)))) <= 0.0)
		tmp = ((-0.5 * (x ^ -0.5)) + (-0.625 * ((x ^ -0.5) / x))) / -x;
	else
		tmp = (((x + 1.0) - x) / (x * (x + 1.0))) / ((x ^ -0.5) + ((x + 1.0) ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(N[(-0.5 * N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision] + N[(-0.625 * N[(N[Power[x, -0.5], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / (-x)), $MachinePrecision], N[(N[(N[(N[(x + 1.0), $MachinePrecision] - x), $MachinePrecision] / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[x, -0.5], $MachinePrecision] + N[Power[N[(x + 1.0), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 x)) (/.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 x #s(literal 1 binary64))))) < 0.0

   1. Initial program 34.0%

      \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 84.2%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + 0.5 \cdot \sqrt{\frac{1}{x}}\right)}{{x}^{2}}} \]

   4. Taylor expanded in x around -inf 0.0%

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-0.125 \cdot \left(\sqrt{\frac{1}{x}} \cdot {\left(\sqrt{-1}\right)}^{2}\right) - 0.5 \cdot \left(\sqrt{\frac{1}{x}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{x} + 0.5 \cdot \left(\sqrt{\frac{1}{x}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{x}} \]
   5. Step-by-step derivation

      1. mul-1-neg 0.0%

         \[\leadsto \color{blue}{-\frac{-1 \cdot \frac{-0.125 \cdot \left(\sqrt{\frac{1}{x}} \cdot {\left(\sqrt{-1}\right)}^{2}\right) - 0.5 \cdot \left(\sqrt{\frac{1}{x}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{x} + 0.5 \cdot \left(\sqrt{\frac{1}{x}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{x}} \]

      2. distribute-neg-frac2 0.0%

         \[\leadsto \color{blue}{\frac{-1 \cdot \frac{-0.125 \cdot \left(\sqrt{\frac{1}{x}} \cdot {\left(\sqrt{-1}\right)}^{2}\right) - 0.5 \cdot \left(\sqrt{\frac{1}{x}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{x} + 0.5 \cdot \left(\sqrt{\frac{1}{x}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{-x}} \]

   6. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{{x}^{-0.5} \cdot -0.5 - \frac{-{x}^{-0.5}}{x} \cdot -0.625}{-x}} \]

   if 0.0 < (-.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 x)) (/.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 x #s(literal 1 binary64)))))

   1. Initial program 51.9%

      \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. flip-- 51.5%

         \[\leadsto \color{blue}{\frac{\frac{1}{\sqrt{x}} \cdot \frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \cdot \frac{1}{\sqrt{x + 1}}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}} \]

      2. div-inv 51.5%

         \[\leadsto \color{blue}{\left(\frac{1}{\sqrt{x}} \cdot \frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \cdot \frac{1}{\sqrt{x + 1}}\right) \cdot \frac{1}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}} \]

      3. frac-times 52.1%

         \[\leadsto \left(\color{blue}{\frac{1 \cdot 1}{\sqrt{x} \cdot \sqrt{x}}} - \frac{1}{\sqrt{x + 1}} \cdot \frac{1}{\sqrt{x + 1}}\right) \cdot \frac{1}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}} \]

      4. metadata-eval 52.1%

         \[\leadsto \left(\frac{\color{blue}{1}}{\sqrt{x} \cdot \sqrt{x}} - \frac{1}{\sqrt{x + 1}} \cdot \frac{1}{\sqrt{x + 1}}\right) \cdot \frac{1}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}} \]

      5. add-sqr-sqrt 52.1%

         \[\leadsto \left(\frac{1}{\color{blue}{x}} - \frac{1}{\sqrt{x + 1}} \cdot \frac{1}{\sqrt{x + 1}}\right) \cdot \frac{1}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}} \]

      6. frac-times 51.2%

         \[\leadsto \left(\frac{1}{x} - \color{blue}{\frac{1 \cdot 1}{\sqrt{x + 1} \cdot \sqrt{x + 1}}}\right) \cdot \frac{1}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}} \]

      7. metadata-eval 51.2%

         \[\leadsto \left(\frac{1}{x} - \frac{\color{blue}{1}}{\sqrt{x + 1} \cdot \sqrt{x + 1}}\right) \cdot \frac{1}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}} \]

      8. add-sqr-sqrt 53.9%

         \[\leadsto \left(\frac{1}{x} - \frac{1}{\color{blue}{x + 1}}\right) \cdot \frac{1}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}} \]

      9. +-commutative 53.9%

         \[\leadsto \left(\frac{1}{x} - \frac{1}{\color{blue}{1 + x}}\right) \cdot \frac{1}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}} \]

      10. inv-pow 53.9%

          \[\leadsto \left(\frac{1}{x} - \frac{1}{1 + x}\right) \cdot \frac{1}{\color{blue}{{\left(\sqrt{x}\right)}^{-1}} + \frac{1}{\sqrt{x + 1}}} \]

      11. sqrt-pow2 53.9%

          \[\leadsto \left(\frac{1}{x} - \frac{1}{1 + x}\right) \cdot \frac{1}{\color{blue}{{x}^{\left(\frac{-1}{2}\right)}} + \frac{1}{\sqrt{x + 1}}} \]

      12. metadata-eval 53.9%

          \[\leadsto \left(\frac{1}{x} - \frac{1}{1 + x}\right) \cdot \frac{1}{{x}^{\color{blue}{-0.5}} + \frac{1}{\sqrt{x + 1}}} \]

      13. pow1/2 53.9%

          \[\leadsto \left(\frac{1}{x} - \frac{1}{1 + x}\right) \cdot \frac{1}{{x}^{-0.5} + \frac{1}{\color{blue}{{\left(x + 1\right)}^{0.5}}}} \]

      14. pow-flip 53.9%

          \[\leadsto \left(\frac{1}{x} - \frac{1}{1 + x}\right) \cdot \frac{1}{{x}^{-0.5} + \color{blue}{{\left(x + 1\right)}^{\left(-0.5\right)}}} \]

      15. +-commutative 53.9%

          \[\leadsto \left(\frac{1}{x} - \frac{1}{1 + x}\right) \cdot \frac{1}{{x}^{-0.5} + {\color{blue}{\left(1 + x\right)}}^{\left(-0.5\right)}} \]

      16. metadata-eval 53.9%

          \[\leadsto \left(\frac{1}{x} - \frac{1}{1 + x}\right) \cdot \frac{1}{{x}^{-0.5} + {\left(1 + x\right)}^{\color{blue}{-0.5}}} \]

   4. Applied egg-rr 53.9%

      \[\leadsto \color{blue}{\left(\frac{1}{x} - \frac{1}{1 + x}\right) \cdot \frac{1}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}}} \]
   5. Step-by-step derivation

      1. associate-*r/ 53.9%

         \[\leadsto \color{blue}{\frac{\left(\frac{1}{x} - \frac{1}{1 + x}\right) \cdot 1}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}}} \]

      2. *-rgt-identity 53.9%

         \[\leadsto \frac{\color{blue}{\frac{1}{x} - \frac{1}{1 + x}}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}} \]

   6. Simplified 53.9%

      \[\leadsto \color{blue}{\frac{\frac{1}{x} - \frac{1}{1 + x}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}}} \]
   7. Step-by-step derivation

      1. frac-sub 99.4%

         \[\leadsto \frac{\color{blue}{\frac{1 \cdot \left(1 + x\right) - x \cdot 1}{x \cdot \left(1 + x\right)}}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}} \]

      2. *-un-lft-identity 99.4%

         \[\leadsto \frac{\frac{\color{blue}{\left(1 + x\right)} - x \cdot 1}{x \cdot \left(1 + x\right)}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}} \]

   8. Applied egg-rr 99.4%

      \[\leadsto \frac{\color{blue}{\frac{\left(1 + x\right) - x \cdot 1}{x \cdot \left(1 + x\right)}}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{x + 1}} \leq 0:\\ \;\;\;\;\frac{-0.5 \cdot {x}^{-0.5} + -0.625 \cdot \frac{{x}^{-0.5}}{x}}{-x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(x + 1\right) - x}{x \cdot \left(x + 1\right)}}{{x}^{-0.5} + {\left(x + 1\right)}^{-0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\left(0.5 \cdot \mathsf{fma}\left(x, 0.25, 1\right)\right) \cdot {x}^{-1.5} - \mathsf{fma}\left(0.5, {x}^{-0.5}, -0.5 \cdot \sqrt{x}\right)}{x}}{x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (/
   (-
    (* (* 0.5 (fma x 0.25 1.0)) (pow x -1.5))
    (fma 0.5 (pow x -0.5) (* -0.5 (sqrt x))))
   x)
  x))

C

double code(double x) {
	return ((((0.5 * fma(x, 0.25, 1.0)) * pow(x, -1.5)) - fma(0.5, pow(x, -0.5), (-0.5 * sqrt(x)))) / x) / x;
}

Julia

function code(x)
	return Float64(Float64(Float64(Float64(Float64(0.5 * fma(x, 0.25, 1.0)) * (x ^ -1.5)) - fma(0.5, (x ^ -0.5), Float64(-0.5 * sqrt(x)))) / x) / x)
end

Wolfram

code[x_] := N[(N[(N[(N[(N[(0.5 * N[(x * 0.25 + 1.0), $MachinePrecision]), $MachinePrecision] * N[Power[x, -1.5], $MachinePrecision]), $MachinePrecision] - N[(0.5 * N[Power[x, -0.5], $MachinePrecision] + N[(-0.5 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 35.0%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
2. Add Preprocessing

3. Taylor expanded in x around inf 84.3%

   \[\leadsto \color{blue}{\frac{0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + 0.5 \cdot \sqrt{\frac{1}{x}}\right)}{{x}^{2}}} \]
4. Step-by-step derivation

   1. *-un-lft-identity 84.3%

      \[\leadsto \frac{\color{blue}{1 \cdot \left(0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}}{{x}^{2}} \]

   2. unpow2 84.3%

      \[\leadsto \frac{1 \cdot \left(0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{\color{blue}{x \cdot x}} \]

   3. times-frac 98.8%

      \[\leadsto \color{blue}{\frac{1}{x} \cdot \frac{0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + 0.5 \cdot \sqrt{\frac{1}{x}}\right)}{x}} \]

5. Applied egg-rr 98.8%

   \[\leadsto \color{blue}{\frac{1}{x} \cdot \frac{0.5 \cdot \left({x}^{-1.5} \cdot \mathsf{fma}\left(x, 0.25, 1\right)\right) - \mathsf{fma}\left(0.5, {x}^{-0.5}, -0.5 \cdot \sqrt{x}\right)}{x}} \]
6. Step-by-step derivation

   1. associate-*l/ 98.9%

      \[\leadsto \color{blue}{\frac{1 \cdot \frac{0.5 \cdot \left({x}^{-1.5} \cdot \mathsf{fma}\left(x, 0.25, 1\right)\right) - \mathsf{fma}\left(0.5, {x}^{-0.5}, -0.5 \cdot \sqrt{x}\right)}{x}}{x}} \]

   2. *-lft-identity 98.9%

      \[\leadsto \frac{\color{blue}{\frac{0.5 \cdot \left({x}^{-1.5} \cdot \mathsf{fma}\left(x, 0.25, 1\right)\right) - \mathsf{fma}\left(0.5, {x}^{-0.5}, -0.5 \cdot \sqrt{x}\right)}{x}}}{x} \]

   3. *-commutative 98.9%

      \[\leadsto \frac{\frac{0.5 \cdot \color{blue}{\left(\mathsf{fma}\left(x, 0.25, 1\right) \cdot {x}^{-1.5}\right)} - \mathsf{fma}\left(0.5, {x}^{-0.5}, -0.5 \cdot \sqrt{x}\right)}{x}}{x} \]

   4. fma-undefine 98.9%

      \[\leadsto \frac{\frac{0.5 \cdot \left(\color{blue}{\left(x \cdot 0.25 + 1\right)} \cdot {x}^{-1.5}\right) - \mathsf{fma}\left(0.5, {x}^{-0.5}, -0.5 \cdot \sqrt{x}\right)}{x}}{x} \]

   5. *-commutative 98.9%

      \[\leadsto \frac{\frac{0.5 \cdot \left(\left(\color{blue}{0.25 \cdot x} + 1\right) \cdot {x}^{-1.5}\right) - \mathsf{fma}\left(0.5, {x}^{-0.5}, -0.5 \cdot \sqrt{x}\right)}{x}}{x} \]

   6. +-commutative 98.9%

      \[\leadsto \frac{\frac{0.5 \cdot \left(\color{blue}{\left(1 + 0.25 \cdot x\right)} \cdot {x}^{-1.5}\right) - \mathsf{fma}\left(0.5, {x}^{-0.5}, -0.5 \cdot \sqrt{x}\right)}{x}}{x} \]

   7. associate-*r* 98.9%

      \[\leadsto \frac{\frac{\color{blue}{\left(0.5 \cdot \left(1 + 0.25 \cdot x\right)\right) \cdot {x}^{-1.5}} - \mathsf{fma}\left(0.5, {x}^{-0.5}, -0.5 \cdot \sqrt{x}\right)}{x}}{x} \]

   8. +-commutative 98.9%

      \[\leadsto \frac{\frac{\left(0.5 \cdot \color{blue}{\left(0.25 \cdot x + 1\right)}\right) \cdot {x}^{-1.5} - \mathsf{fma}\left(0.5, {x}^{-0.5}, -0.5 \cdot \sqrt{x}\right)}{x}}{x} \]

   9. *-commutative 98.9%

      \[\leadsto \frac{\frac{\left(0.5 \cdot \left(\color{blue}{x \cdot 0.25} + 1\right)\right) \cdot {x}^{-1.5} - \mathsf{fma}\left(0.5, {x}^{-0.5}, -0.5 \cdot \sqrt{x}\right)}{x}}{x} \]

   10. fma-undefine 98.9%

       \[\leadsto \frac{\frac{\left(0.5 \cdot \color{blue}{\mathsf{fma}\left(x, 0.25, 1\right)}\right) \cdot {x}^{-1.5} - \mathsf{fma}\left(0.5, {x}^{-0.5}, -0.5 \cdot \sqrt{x}\right)}{x}}{x} \]

   11. *-commutative 98.9%

       \[\leadsto \frac{\frac{\left(0.5 \cdot \mathsf{fma}\left(x, 0.25, 1\right)\right) \cdot {x}^{-1.5} - \mathsf{fma}\left(0.5, {x}^{-0.5}, \color{blue}{\sqrt{x} \cdot -0.5}\right)}{x}}{x} \]

7. Simplified 98.9%

   \[\leadsto \color{blue}{\frac{\frac{\left(0.5 \cdot \mathsf{fma}\left(x, 0.25, 1\right)\right) \cdot {x}^{-1.5} - \mathsf{fma}\left(0.5, {x}^{-0.5}, \sqrt{x} \cdot -0.5\right)}{x}}{x}} \]

8. Final simplification 98.9%

   \[\leadsto \frac{\frac{\left(0.5 \cdot \mathsf{fma}\left(x, 0.25, 1\right)\right) \cdot {x}^{-1.5} - \mathsf{fma}\left(0.5, {x}^{-0.5}, -0.5 \cdot \sqrt{x}\right)}{x}}{x} \]
9. Add Preprocessing

Alternative 3: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{x + 1}} \leq 0:\\ \;\;\;\;\frac{-0.5 \cdot {x}^{-0.5} + -0.625 \cdot \frac{{x}^{-0.5}}{x}}{-x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{x}}{x + 1}}{{x}^{-0.5} + {\left(x + 1\right)}^{-0.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (/ 1.0 (sqrt x)) (/ -1.0 (sqrt (+ x 1.0)))) 0.0)
   (/ (+ (* -0.5 (pow x -0.5)) (* -0.625 (/ (pow x -0.5) x))) (- x))
   (/ (/ (/ 1.0 x) (+ x 1.0)) (+ (pow x -0.5) (pow (+ x 1.0) -0.5)))))

C

double code(double x) {
	double tmp;
	if (((1.0 / sqrt(x)) + (-1.0 / sqrt((x + 1.0)))) <= 0.0) {
		tmp = ((-0.5 * pow(x, -0.5)) + (-0.625 * (pow(x, -0.5) / x))) / -x;
	} else {
		tmp = ((1.0 / x) / (x + 1.0)) / (pow(x, -0.5) + pow((x + 1.0), -0.5));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (((1.0d0 / sqrt(x)) + ((-1.0d0) / sqrt((x + 1.0d0)))) <= 0.0d0) then
        tmp = (((-0.5d0) * (x ** (-0.5d0))) + ((-0.625d0) * ((x ** (-0.5d0)) / x))) / -x
    else
        tmp = ((1.0d0 / x) / (x + 1.0d0)) / ((x ** (-0.5d0)) + ((x + 1.0d0) ** (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (((1.0 / Math.sqrt(x)) + (-1.0 / Math.sqrt((x + 1.0)))) <= 0.0) {
		tmp = ((-0.5 * Math.pow(x, -0.5)) + (-0.625 * (Math.pow(x, -0.5) / x))) / -x;
	} else {
		tmp = ((1.0 / x) / (x + 1.0)) / (Math.pow(x, -0.5) + Math.pow((x + 1.0), -0.5));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if ((1.0 / math.sqrt(x)) + (-1.0 / math.sqrt((x + 1.0)))) <= 0.0:
		tmp = ((-0.5 * math.pow(x, -0.5)) + (-0.625 * (math.pow(x, -0.5) / x))) / -x
	else:
		tmp = ((1.0 / x) / (x + 1.0)) / (math.pow(x, -0.5) + math.pow((x + 1.0), -0.5))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(x)) + Float64(-1.0 / sqrt(Float64(x + 1.0)))) <= 0.0)
		tmp = Float64(Float64(Float64(-0.5 * (x ^ -0.5)) + Float64(-0.625 * Float64((x ^ -0.5) / x))) / Float64(-x));
	else
		tmp = Float64(Float64(Float64(1.0 / x) / Float64(x + 1.0)) / Float64((x ^ -0.5) + (Float64(x + 1.0) ^ -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (((1.0 / sqrt(x)) + (-1.0 / sqrt((x + 1.0)))) <= 0.0)
		tmp = ((-0.5 * (x ^ -0.5)) + (-0.625 * ((x ^ -0.5) / x))) / -x;
	else
		tmp = ((1.0 / x) / (x + 1.0)) / ((x ^ -0.5) + ((x + 1.0) ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(N[(-0.5 * N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision] + N[(-0.625 * N[(N[Power[x, -0.5], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / (-x)), $MachinePrecision], N[(N[(N[(1.0 / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / N[(N[Power[x, -0.5], $MachinePrecision] + N[Power[N[(x + 1.0), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 x)) (/.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 x #s(literal 1 binary64))))) < 0.0

   1. Initial program 34.0%

      \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 84.2%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + 0.5 \cdot \sqrt{\frac{1}{x}}\right)}{{x}^{2}}} \]

   4. Taylor expanded in x around -inf 0.0%

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-0.125 \cdot \left(\sqrt{\frac{1}{x}} \cdot {\left(\sqrt{-1}\right)}^{2}\right) - 0.5 \cdot \left(\sqrt{\frac{1}{x}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{x} + 0.5 \cdot \left(\sqrt{\frac{1}{x}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{x}} \]
   5. Step-by-step derivation

      1. mul-1-neg 0.0%

         \[\leadsto \color{blue}{-\frac{-1 \cdot \frac{-0.125 \cdot \left(\sqrt{\frac{1}{x}} \cdot {\left(\sqrt{-1}\right)}^{2}\right) - 0.5 \cdot \left(\sqrt{\frac{1}{x}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{x} + 0.5 \cdot \left(\sqrt{\frac{1}{x}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{x}} \]

      2. distribute-neg-frac2 0.0%

         \[\leadsto \color{blue}{\frac{-1 \cdot \frac{-0.125 \cdot \left(\sqrt{\frac{1}{x}} \cdot {\left(\sqrt{-1}\right)}^{2}\right) - 0.5 \cdot \left(\sqrt{\frac{1}{x}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{x} + 0.5 \cdot \left(\sqrt{\frac{1}{x}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{-x}} \]

   6. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{{x}^{-0.5} \cdot -0.5 - \frac{-{x}^{-0.5}}{x} \cdot -0.625}{-x}} \]

   if 0.0 < (-.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 x)) (/.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 x #s(literal 1 binary64)))))

   1. Initial program 51.9%

      \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. flip-- 51.5%

         \[\leadsto \color{blue}{\frac{\frac{1}{\sqrt{x}} \cdot \frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \cdot \frac{1}{\sqrt{x + 1}}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}} \]

      2. div-inv 51.5%

         \[\leadsto \color{blue}{\left(\frac{1}{\sqrt{x}} \cdot \frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \cdot \frac{1}{\sqrt{x + 1}}\right) \cdot \frac{1}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}} \]

      3. frac-times 52.1%

         \[\leadsto \left(\color{blue}{\frac{1 \cdot 1}{\sqrt{x} \cdot \sqrt{x}}} - \frac{1}{\sqrt{x + 1}} \cdot \frac{1}{\sqrt{x + 1}}\right) \cdot \frac{1}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}} \]

      4. metadata-eval 52.1%

         \[\leadsto \left(\frac{\color{blue}{1}}{\sqrt{x} \cdot \sqrt{x}} - \frac{1}{\sqrt{x + 1}} \cdot \frac{1}{\sqrt{x + 1}}\right) \cdot \frac{1}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}} \]

      5. add-sqr-sqrt 52.1%

         \[\leadsto \left(\frac{1}{\color{blue}{x}} - \frac{1}{\sqrt{x + 1}} \cdot \frac{1}{\sqrt{x + 1}}\right) \cdot \frac{1}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}} \]

      6. frac-times 51.2%

         \[\leadsto \left(\frac{1}{x} - \color{blue}{\frac{1 \cdot 1}{\sqrt{x + 1} \cdot \sqrt{x + 1}}}\right) \cdot \frac{1}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}} \]

      7. metadata-eval 51.2%

         \[\leadsto \left(\frac{1}{x} - \frac{\color{blue}{1}}{\sqrt{x + 1} \cdot \sqrt{x + 1}}\right) \cdot \frac{1}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}} \]

      8. add-sqr-sqrt 53.9%

         \[\leadsto \left(\frac{1}{x} - \frac{1}{\color{blue}{x + 1}}\right) \cdot \frac{1}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}} \]

      9. +-commutative 53.9%

         \[\leadsto \left(\frac{1}{x} - \frac{1}{\color{blue}{1 + x}}\right) \cdot \frac{1}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}} \]

      10. inv-pow 53.9%

          \[\leadsto \left(\frac{1}{x} - \frac{1}{1 + x}\right) \cdot \frac{1}{\color{blue}{{\left(\sqrt{x}\right)}^{-1}} + \frac{1}{\sqrt{x + 1}}} \]

      11. sqrt-pow2 53.9%

          \[\leadsto \left(\frac{1}{x} - \frac{1}{1 + x}\right) \cdot \frac{1}{\color{blue}{{x}^{\left(\frac{-1}{2}\right)}} + \frac{1}{\sqrt{x + 1}}} \]

      12. metadata-eval 53.9%

          \[\leadsto \left(\frac{1}{x} - \frac{1}{1 + x}\right) \cdot \frac{1}{{x}^{\color{blue}{-0.5}} + \frac{1}{\sqrt{x + 1}}} \]

      13. pow1/2 53.9%

          \[\leadsto \left(\frac{1}{x} - \frac{1}{1 + x}\right) \cdot \frac{1}{{x}^{-0.5} + \frac{1}{\color{blue}{{\left(x + 1\right)}^{0.5}}}} \]

      14. pow-flip 53.9%

          \[\leadsto \left(\frac{1}{x} - \frac{1}{1 + x}\right) \cdot \frac{1}{{x}^{-0.5} + \color{blue}{{\left(x + 1\right)}^{\left(-0.5\right)}}} \]

      15. +-commutative 53.9%

          \[\leadsto \left(\frac{1}{x} - \frac{1}{1 + x}\right) \cdot \frac{1}{{x}^{-0.5} + {\color{blue}{\left(1 + x\right)}}^{\left(-0.5\right)}} \]

      16. metadata-eval 53.9%

          \[\leadsto \left(\frac{1}{x} - \frac{1}{1 + x}\right) \cdot \frac{1}{{x}^{-0.5} + {\left(1 + x\right)}^{\color{blue}{-0.5}}} \]

   4. Applied egg-rr 53.9%

      \[\leadsto \color{blue}{\left(\frac{1}{x} - \frac{1}{1 + x}\right) \cdot \frac{1}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}}} \]
   5. Step-by-step derivation

      1. associate-*r/ 53.9%

         \[\leadsto \color{blue}{\frac{\left(\frac{1}{x} - \frac{1}{1 + x}\right) \cdot 1}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}}} \]

      2. *-rgt-identity 53.9%

         \[\leadsto \frac{\color{blue}{\frac{1}{x} - \frac{1}{1 + x}}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}} \]

   6. Simplified 53.9%

      \[\leadsto \color{blue}{\frac{\frac{1}{x} - \frac{1}{1 + x}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}}} \]
   7. Step-by-step derivation

      1. frac-sub 99.4%

         \[\leadsto \frac{\color{blue}{\frac{1 \cdot \left(1 + x\right) - x \cdot 1}{x \cdot \left(1 + x\right)}}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}} \]

      2. *-un-lft-identity 99.4%

         \[\leadsto \frac{\frac{\color{blue}{\left(1 + x\right)} - x \cdot 1}{x \cdot \left(1 + x\right)}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}} \]

   8. Applied egg-rr 99.4%

      \[\leadsto \frac{\color{blue}{\frac{\left(1 + x\right) - x \cdot 1}{x \cdot \left(1 + x\right)}}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}} \]
   9. Step-by-step derivation

      1. /-rgt-identity 99.4%

         \[\leadsto \frac{\frac{\color{blue}{\frac{1 + x}{1}} - x \cdot 1}{x \cdot \left(1 + x\right)}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}} \]

      2. *-lft-identity 99.4%

         \[\leadsto \frac{\frac{\color{blue}{1 \cdot \frac{1 + x}{1}} - x \cdot 1}{x \cdot \left(1 + x\right)}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}} \]

      3. /-rgt-identity 99.4%

         \[\leadsto \frac{\frac{1 \cdot \frac{1 + x}{1} - \color{blue}{\frac{x}{1}} \cdot 1}{x \cdot \left(1 + x\right)}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}} \]

      4. *-lft-identity 99.4%

         \[\leadsto \frac{\frac{\color{blue}{\frac{1 + x}{1}} - \frac{x}{1} \cdot 1}{x \cdot \left(1 + x\right)}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}} \]

      5. /-rgt-identity 99.4%

         \[\leadsto \frac{\frac{\color{blue}{\left(1 + x\right)} - \frac{x}{1} \cdot 1}{x \cdot \left(1 + x\right)}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}} \]

      6. *-rgt-identity 99.4%

         \[\leadsto \frac{\frac{\left(1 + x\right) - \color{blue}{\frac{x}{1}}}{x \cdot \left(1 + x\right)}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}} \]

      7. /-rgt-identity 99.4%

         \[\leadsto \frac{\frac{\left(1 + x\right) - \color{blue}{x}}{x \cdot \left(1 + x\right)}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}} \]

      8. associate--l+ 99.4%

         \[\leadsto \frac{\frac{\color{blue}{1 + \left(x - x\right)}}{x \cdot \left(1 + x\right)}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}} \]

      9. +-inverses 99.4%

         \[\leadsto \frac{\frac{1 + \color{blue}{0}}{x \cdot \left(1 + x\right)}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}} \]

      10. metadata-eval 99.4%

          \[\leadsto \frac{\frac{\color{blue}{1}}{x \cdot \left(1 + x\right)}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}} \]

      11. associate-/r* 99.3%

          \[\leadsto \frac{\color{blue}{\frac{\frac{1}{x}}{1 + x}}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}} \]

   10. Simplified 99.3%

       \[\leadsto \frac{\color{blue}{\frac{\frac{1}{x}}{1 + x}}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{x + 1}} \leq 0:\\ \;\;\;\;\frac{-0.5 \cdot {x}^{-0.5} + -0.625 \cdot \frac{{x}^{-0.5}}{x}}{-x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{x}}{x + 1}}{{x}^{-0.5} + {\left(x + 1\right)}^{-0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.125 \cdot \sqrt{\frac{1}{x}} - \mathsf{fma}\left(0.5, {x}^{-0.5}, -0.5 \cdot \sqrt{x}\right)}{x}}{x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (/ (- (* 0.125 (sqrt (/ 1.0 x))) (fma 0.5 (pow x -0.5) (* -0.5 (sqrt x)))) x)
  x))

C

double code(double x) {
	return (((0.125 * sqrt((1.0 / x))) - fma(0.5, pow(x, -0.5), (-0.5 * sqrt(x)))) / x) / x;
}

Julia

function code(x)
	return Float64(Float64(Float64(Float64(0.125 * sqrt(Float64(1.0 / x))) - fma(0.5, (x ^ -0.5), Float64(-0.5 * sqrt(x)))) / x) / x)
end

Wolfram

code[x_] := N[(N[(N[(N[(0.125 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(0.5 * N[Power[x, -0.5], $MachinePrecision] + N[(-0.5 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 35.0%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
2. Add Preprocessing

3. Taylor expanded in x around inf 84.3%

   \[\leadsto \color{blue}{\frac{0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + 0.5 \cdot \sqrt{\frac{1}{x}}\right)}{{x}^{2}}} \]
4. Step-by-step derivation

   1. *-un-lft-identity 84.3%

      \[\leadsto \frac{\color{blue}{1 \cdot \left(0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}}{{x}^{2}} \]

   2. unpow2 84.3%

      \[\leadsto \frac{1 \cdot \left(0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{\color{blue}{x \cdot x}} \]

   3. times-frac 98.8%

      \[\leadsto \color{blue}{\frac{1}{x} \cdot \frac{0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + 0.5 \cdot \sqrt{\frac{1}{x}}\right)}{x}} \]

5. Applied egg-rr 98.8%

   \[\leadsto \color{blue}{\frac{1}{x} \cdot \frac{0.5 \cdot \left({x}^{-1.5} \cdot \mathsf{fma}\left(x, 0.25, 1\right)\right) - \mathsf{fma}\left(0.5, {x}^{-0.5}, -0.5 \cdot \sqrt{x}\right)}{x}} \]
6. Step-by-step derivation

   1. associate-*l/ 98.9%

      \[\leadsto \color{blue}{\frac{1 \cdot \frac{0.5 \cdot \left({x}^{-1.5} \cdot \mathsf{fma}\left(x, 0.25, 1\right)\right) - \mathsf{fma}\left(0.5, {x}^{-0.5}, -0.5 \cdot \sqrt{x}\right)}{x}}{x}} \]

   2. *-lft-identity 98.9%

      \[\leadsto \frac{\color{blue}{\frac{0.5 \cdot \left({x}^{-1.5} \cdot \mathsf{fma}\left(x, 0.25, 1\right)\right) - \mathsf{fma}\left(0.5, {x}^{-0.5}, -0.5 \cdot \sqrt{x}\right)}{x}}}{x} \]

   3. *-commutative 98.9%

      \[\leadsto \frac{\frac{0.5 \cdot \color{blue}{\left(\mathsf{fma}\left(x, 0.25, 1\right) \cdot {x}^{-1.5}\right)} - \mathsf{fma}\left(0.5, {x}^{-0.5}, -0.5 \cdot \sqrt{x}\right)}{x}}{x} \]

   4. fma-undefine 98.9%

      \[\leadsto \frac{\frac{0.5 \cdot \left(\color{blue}{\left(x \cdot 0.25 + 1\right)} \cdot {x}^{-1.5}\right) - \mathsf{fma}\left(0.5, {x}^{-0.5}, -0.5 \cdot \sqrt{x}\right)}{x}}{x} \]

   5. *-commutative 98.9%

      \[\leadsto \frac{\frac{0.5 \cdot \left(\left(\color{blue}{0.25 \cdot x} + 1\right) \cdot {x}^{-1.5}\right) - \mathsf{fma}\left(0.5, {x}^{-0.5}, -0.5 \cdot \sqrt{x}\right)}{x}}{x} \]

   6. +-commutative 98.9%

      \[\leadsto \frac{\frac{0.5 \cdot \left(\color{blue}{\left(1 + 0.25 \cdot x\right)} \cdot {x}^{-1.5}\right) - \mathsf{fma}\left(0.5, {x}^{-0.5}, -0.5 \cdot \sqrt{x}\right)}{x}}{x} \]

   7. associate-*r* 98.9%

      \[\leadsto \frac{\frac{\color{blue}{\left(0.5 \cdot \left(1 + 0.25 \cdot x\right)\right) \cdot {x}^{-1.5}} - \mathsf{fma}\left(0.5, {x}^{-0.5}, -0.5 \cdot \sqrt{x}\right)}{x}}{x} \]

   8. +-commutative 98.9%

      \[\leadsto \frac{\frac{\left(0.5 \cdot \color{blue}{\left(0.25 \cdot x + 1\right)}\right) \cdot {x}^{-1.5} - \mathsf{fma}\left(0.5, {x}^{-0.5}, -0.5 \cdot \sqrt{x}\right)}{x}}{x} \]

   9. *-commutative 98.9%

      \[\leadsto \frac{\frac{\left(0.5 \cdot \left(\color{blue}{x \cdot 0.25} + 1\right)\right) \cdot {x}^{-1.5} - \mathsf{fma}\left(0.5, {x}^{-0.5}, -0.5 \cdot \sqrt{x}\right)}{x}}{x} \]

   10. fma-undefine 98.9%

       \[\leadsto \frac{\frac{\left(0.5 \cdot \color{blue}{\mathsf{fma}\left(x, 0.25, 1\right)}\right) \cdot {x}^{-1.5} - \mathsf{fma}\left(0.5, {x}^{-0.5}, -0.5 \cdot \sqrt{x}\right)}{x}}{x} \]

   11. *-commutative 98.9%

       \[\leadsto \frac{\frac{\left(0.5 \cdot \mathsf{fma}\left(x, 0.25, 1\right)\right) \cdot {x}^{-1.5} - \mathsf{fma}\left(0.5, {x}^{-0.5}, \color{blue}{\sqrt{x} \cdot -0.5}\right)}{x}}{x} \]

7. Simplified 98.9%

   \[\leadsto \color{blue}{\frac{\frac{\left(0.5 \cdot \mathsf{fma}\left(x, 0.25, 1\right)\right) \cdot {x}^{-1.5} - \mathsf{fma}\left(0.5, {x}^{-0.5}, \sqrt{x} \cdot -0.5\right)}{x}}{x}} \]

8. Taylor expanded in x around inf 98.9%

   \[\leadsto \frac{\frac{\color{blue}{0.125 \cdot \sqrt{\frac{1}{x}}} - \mathsf{fma}\left(0.5, {x}^{-0.5}, \sqrt{x} \cdot -0.5\right)}{x}}{x} \]

9. Final simplification 98.9%

   \[\leadsto \frac{\frac{0.125 \cdot \sqrt{\frac{1}{x}} - \mathsf{fma}\left(0.5, {x}^{-0.5}, -0.5 \cdot \sqrt{x}\right)}{x}}{x} \]
10. Add Preprocessing

Alternative 5: 97.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.5 \cdot \sqrt{\frac{1}{x}}}{x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (* 0.5 (sqrt (/ 1.0 x))) x))

C

double code(double x) {
	return (0.5 * sqrt((1.0 / x))) / x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (0.5d0 * sqrt((1.0d0 / x))) / x
end function

Java

public static double code(double x) {
	return (0.5 * Math.sqrt((1.0 / x))) / x;
}

Python

def code(x):
	return (0.5 * math.sqrt((1.0 / x))) / x

Julia

function code(x)
	return Float64(Float64(0.5 * sqrt(Float64(1.0 / x))) / x)
end

MATLAB

function tmp = code(x)
	tmp = (0.5 * sqrt((1.0 / x))) / x;
end

Wolfram

code[x_] := N[(N[(0.5 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 35.0%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
2. Add Preprocessing

3. Taylor expanded in x around inf 84.3%

   \[\leadsto \color{blue}{\frac{0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + 0.5 \cdot \sqrt{\frac{1}{x}}\right)}{{x}^{2}}} \]
4. Step-by-step derivation

   1. *-un-lft-identity 84.3%

      \[\leadsto \frac{\color{blue}{1 \cdot \left(0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}}{{x}^{2}} \]

   2. unpow2 84.3%

      \[\leadsto \frac{1 \cdot \left(0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{\color{blue}{x \cdot x}} \]

   3. times-frac 98.8%

      \[\leadsto \color{blue}{\frac{1}{x} \cdot \frac{0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + 0.5 \cdot \sqrt{\frac{1}{x}}\right)}{x}} \]

5. Applied egg-rr 98.8%

   \[\leadsto \color{blue}{\frac{1}{x} \cdot \frac{0.5 \cdot \left({x}^{-1.5} \cdot \mathsf{fma}\left(x, 0.25, 1\right)\right) - \mathsf{fma}\left(0.5, {x}^{-0.5}, -0.5 \cdot \sqrt{x}\right)}{x}} \]
6. Step-by-step derivation

   1. associate-*l/ 98.9%

      \[\leadsto \color{blue}{\frac{1 \cdot \frac{0.5 \cdot \left({x}^{-1.5} \cdot \mathsf{fma}\left(x, 0.25, 1\right)\right) - \mathsf{fma}\left(0.5, {x}^{-0.5}, -0.5 \cdot \sqrt{x}\right)}{x}}{x}} \]

   2. *-lft-identity 98.9%

      \[\leadsto \frac{\color{blue}{\frac{0.5 \cdot \left({x}^{-1.5} \cdot \mathsf{fma}\left(x, 0.25, 1\right)\right) - \mathsf{fma}\left(0.5, {x}^{-0.5}, -0.5 \cdot \sqrt{x}\right)}{x}}}{x} \]

   3. *-commutative 98.9%

      \[\leadsto \frac{\frac{0.5 \cdot \color{blue}{\left(\mathsf{fma}\left(x, 0.25, 1\right) \cdot {x}^{-1.5}\right)} - \mathsf{fma}\left(0.5, {x}^{-0.5}, -0.5 \cdot \sqrt{x}\right)}{x}}{x} \]

   4. fma-undefine 98.9%

      \[\leadsto \frac{\frac{0.5 \cdot \left(\color{blue}{\left(x \cdot 0.25 + 1\right)} \cdot {x}^{-1.5}\right) - \mathsf{fma}\left(0.5, {x}^{-0.5}, -0.5 \cdot \sqrt{x}\right)}{x}}{x} \]

   5. *-commutative 98.9%

      \[\leadsto \frac{\frac{0.5 \cdot \left(\left(\color{blue}{0.25 \cdot x} + 1\right) \cdot {x}^{-1.5}\right) - \mathsf{fma}\left(0.5, {x}^{-0.5}, -0.5 \cdot \sqrt{x}\right)}{x}}{x} \]

   6. +-commutative 98.9%

      \[\leadsto \frac{\frac{0.5 \cdot \left(\color{blue}{\left(1 + 0.25 \cdot x\right)} \cdot {x}^{-1.5}\right) - \mathsf{fma}\left(0.5, {x}^{-0.5}, -0.5 \cdot \sqrt{x}\right)}{x}}{x} \]

   7. associate-*r* 98.9%

      \[\leadsto \frac{\frac{\color{blue}{\left(0.5 \cdot \left(1 + 0.25 \cdot x\right)\right) \cdot {x}^{-1.5}} - \mathsf{fma}\left(0.5, {x}^{-0.5}, -0.5 \cdot \sqrt{x}\right)}{x}}{x} \]

   8. +-commutative 98.9%

      \[\leadsto \frac{\frac{\left(0.5 \cdot \color{blue}{\left(0.25 \cdot x + 1\right)}\right) \cdot {x}^{-1.5} - \mathsf{fma}\left(0.5, {x}^{-0.5}, -0.5 \cdot \sqrt{x}\right)}{x}}{x} \]

   9. *-commutative 98.9%

      \[\leadsto \frac{\frac{\left(0.5 \cdot \left(\color{blue}{x \cdot 0.25} + 1\right)\right) \cdot {x}^{-1.5} - \mathsf{fma}\left(0.5, {x}^{-0.5}, -0.5 \cdot \sqrt{x}\right)}{x}}{x} \]

   10. fma-undefine 98.9%

       \[\leadsto \frac{\frac{\left(0.5 \cdot \color{blue}{\mathsf{fma}\left(x, 0.25, 1\right)}\right) \cdot {x}^{-1.5} - \mathsf{fma}\left(0.5, {x}^{-0.5}, -0.5 \cdot \sqrt{x}\right)}{x}}{x} \]

   11. *-commutative 98.9%

       \[\leadsto \frac{\frac{\left(0.5 \cdot \mathsf{fma}\left(x, 0.25, 1\right)\right) \cdot {x}^{-1.5} - \mathsf{fma}\left(0.5, {x}^{-0.5}, \color{blue}{\sqrt{x} \cdot -0.5}\right)}{x}}{x} \]

7. Simplified 98.9%

   \[\leadsto \color{blue}{\frac{\frac{\left(0.5 \cdot \mathsf{fma}\left(x, 0.25, 1\right)\right) \cdot {x}^{-1.5} - \mathsf{fma}\left(0.5, {x}^{-0.5}, \sqrt{x} \cdot -0.5\right)}{x}}{x}} \]

8. Taylor expanded in x around inf 98.0%

   \[\leadsto \frac{\color{blue}{0.5 \cdot \sqrt{\frac{1}{x}}}}{x} \]
9. Add Preprocessing

Alternative 6: 35.1% accurate, 209.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.0d0
end function

Java

public static double code(double x) {
	return 0.0;
}

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

function tmp = code(x)
	tmp = 0.0;
end

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 35.0%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. add-cube-cbrt 10.5%

      \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{1}{\sqrt{x}}} \cdot \sqrt[3]{\frac{1}{\sqrt{x}}}\right) \cdot \sqrt[3]{\frac{1}{\sqrt{x}}}} - \frac{1}{\sqrt{x + 1}} \]

   2. associate-*l* 10.5%

      \[\leadsto \color{blue}{\sqrt[3]{\frac{1}{\sqrt{x}}} \cdot \left(\sqrt[3]{\frac{1}{\sqrt{x}}} \cdot \sqrt[3]{\frac{1}{\sqrt{x}}}\right)} - \frac{1}{\sqrt{x + 1}} \]

   3. frac-2neg 10.5%

      \[\leadsto \sqrt[3]{\frac{1}{\sqrt{x}}} \cdot \left(\sqrt[3]{\frac{1}{\sqrt{x}}} \cdot \sqrt[3]{\frac{1}{\sqrt{x}}}\right) - \color{blue}{\frac{-1}{-\sqrt{x + 1}}} \]

   4. metadata-eval 10.5%

      \[\leadsto \sqrt[3]{\frac{1}{\sqrt{x}}} \cdot \left(\sqrt[3]{\frac{1}{\sqrt{x}}} \cdot \sqrt[3]{\frac{1}{\sqrt{x}}}\right) - \frac{\color{blue}{-1}}{-\sqrt{x + 1}} \]

   5. div-inv 10.5%

      \[\leadsto \sqrt[3]{\frac{1}{\sqrt{x}}} \cdot \left(\sqrt[3]{\frac{1}{\sqrt{x}}} \cdot \sqrt[3]{\frac{1}{\sqrt{x}}}\right) - \color{blue}{-1 \cdot \frac{1}{-\sqrt{x + 1}}} \]

   6. distribute-neg-frac2 10.5%

      \[\leadsto \sqrt[3]{\frac{1}{\sqrt{x}}} \cdot \left(\sqrt[3]{\frac{1}{\sqrt{x}}} \cdot \sqrt[3]{\frac{1}{\sqrt{x}}}\right) - -1 \cdot \color{blue}{\left(-\frac{1}{\sqrt{x + 1}}\right)} \]

   7. prod-diff 6.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt[3]{\frac{1}{\sqrt{x}}}, \sqrt[3]{\frac{1}{\sqrt{x}}} \cdot \sqrt[3]{\frac{1}{\sqrt{x}}}, -\left(-\frac{1}{\sqrt{x + 1}}\right) \cdot -1\right) + \mathsf{fma}\left(-\left(-\frac{1}{\sqrt{x + 1}}\right), -1, \left(-\frac{1}{\sqrt{x + 1}}\right) \cdot -1\right)} \]

4. Applied egg-rr 6.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt[3]{{x}^{-0.5}}, \sqrt[3]{\frac{1}{x}}, -\frac{-1}{\sqrt{1 + x}} \cdot -1\right) + \mathsf{fma}\left({\left(1 + x\right)}^{-0.5}, -1, \frac{-1}{\sqrt{1 + x}} \cdot -1\right)} \]
5. Step-by-step derivation

   1. +-commutative 6.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(1 + x\right)}^{-0.5}, -1, \frac{-1}{\sqrt{1 + x}} \cdot -1\right) + \mathsf{fma}\left(\sqrt[3]{{x}^{-0.5}}, \sqrt[3]{\frac{1}{x}}, -\frac{-1}{\sqrt{1 + x}} \cdot -1\right)} \]

6. Simplified 6.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt[3]{{x}^{-0.5}}, \sqrt[3]{\frac{1}{x}}, \frac{-1}{\sqrt{1 + x}}\right)} \]

7. Taylor expanded in x around inf 32.4%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{x}} + -1 \cdot \sqrt{\frac{1}{x}}} \]
8. Step-by-step derivation

   1. distribute-rgt1-in 32.4%

      \[\leadsto \color{blue}{\left(-1 + 1\right) \cdot \sqrt{\frac{1}{x}}} \]

   2. metadata-eval 32.4%

      \[\leadsto \color{blue}{0} \cdot \sqrt{\frac{1}{x}} \]

   3. mul0-lft 32.4%

      \[\leadsto \color{blue}{0} \]

9. Simplified 32.4%

   \[\leadsto \color{blue}{0} \]
10. Add Preprocessing

Developer target: 98.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ 1.0 (+ (* (+ x 1.0) (sqrt x)) (* x (sqrt (+ x 1.0))))))

C

double code(double x) {
	return 1.0 / (((x + 1.0) * sqrt(x)) + (x * sqrt((x + 1.0))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 / (((x + 1.0d0) * sqrt(x)) + (x * sqrt((x + 1.0d0))))
end function

Java

public static double code(double x) {
	return 1.0 / (((x + 1.0) * Math.sqrt(x)) + (x * Math.sqrt((x + 1.0))));
}

Python

def code(x):
	return 1.0 / (((x + 1.0) * math.sqrt(x)) + (x * math.sqrt((x + 1.0))))

Julia

function code(x)
	return Float64(1.0 / Float64(Float64(Float64(x + 1.0) * sqrt(x)) + Float64(x * sqrt(Float64(x + 1.0)))))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 / (((x + 1.0) * sqrt(x)) + (x * sqrt((x + 1.0))));
end

Wolfram

code[x_] := N[(1.0 / N[(N[(N[(x + 1.0), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(x * N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024086 
(FPCore (x)
  :name "2isqrt (example 3.6)"
  :precision binary64
  :pre (and (> x 1.0) (< x 1e+308))

  :alt
  (/ 1.0 (+ (* (+ x 1.0) (sqrt x)) (* x (sqrt (+ x 1.0)))))

  (- (/ 1.0 (sqrt x)) (/ 1.0 (sqrt (+ x 1.0)))))